Role of dynamic exchange coupling in magnetic relaxations of metallic multilayer films „ invited …

Role of dynamic exchange coupling in magnetic relaxations of metallic multilayer films „ invited …

B. Heinrich, G. Woltersdorf,^a)and R. Urban

Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, V5A 1S6, Canada E. Simanek

6255 Charing Lane, Cambria, California 93428 共Presented on 14 November 2002兲

The relaxation processes were investigated by ferromagnetic resonance 共FMR兲 using magnetic single, Au/Fe/GaAs共001兲, and double layer, Au/Fe/Au/Fe/GaAs共001兲, structures prepared by molecular beam epitaxy. These structures provided an excellent opportunity to investigate nonlocal damping which is caused by spin transport across a nonmagnetic spacer. In the double layer structures thin Fe layers F1 were separated from a second thick Fe layer F2 by a Au共001兲, normal metal spacer. The interface magnetic anisotropies separated the FMR fields of F1 and F2 by a big margin which allowed us to investigate FMR in F1 while F2 had a negligible angle of precession.

The main result is that the ultrathin Fe films in magnetic double layers acquire a nonlocal interface Gilbert damping. Several mechanisms have been put forward to explain the nonlocal damping. A brief review of each mechanism will be presented. They will be compared with the experimental results allowing one to critically assess their applicability and strength. It will be shown that the precessing layers act as spin pumps and spin sinks. This concept was tested by investigating the FMR linewidth around an accidental crossover of the resonance fields for the layers F1 and F2.

© 2003 American Institute of Physics. 关DOI: 10.1063/1.1543852兴

INTRODUCTION

The small lateral dimensions of spintronics devices and high density memory bits require the use of magnetic metal- lic ultrathin film structures where the magnetic moments across the film thickness are locked together by the intra layer exchange coupling. Spintronics and high density mag- netic recording employ fast magnetization reversal pro- cesses. It is currently of considerable interest to acquire a thorough understanding of the spin dynamics and magnetic relaxation processes in the nano-second time regime. The spin dynamics in the classical limit can be described by the Landau–Lifshitz–Gilbert equation of motion

1

␥

⳵^M

⳵^{t ⫽⫺关MÃHeff兴⫹}

G

␥^2Ms

2

冋

册

where␥ is the absolute value of the electron gyromagnetic ratio, M_sis the saturation magnetization and G is the Gilbert damping parameter. The effective field H_eff is given by the derivatives of the Gibbs energy, U, with respect to the com- ponents ( M_x, M_y, M_z) of the magnetization vector M(t), see.¹The second term in Eq.共1兲represents the well known Gilbert damping torque. The purpose of this article is to re- view the basic concepts of magnetic relaxations with empha- sis on metallic multilayers.

NONLOCAL DAMPING, EXPERIMENT

The role of interface damping was investigated in high quality crystalline Au/Fe/Au/Fe共001兲 structures grown on GaAs共001兲substrates.^2,26,27The in-plane ferromagnetic reso- nance共FMR兲experiments were carried out using 10, 24, 36, and 72 GHz systems.³

Single Fe ultrathin films with thicknesses of 8, 11, 16, 21, and 31 monolayers 共MLs兲 were grown directly on GaAs共001兲. They were covered by a 20 ML thick Au共001兲 cap layer for protection in ambient conditions. FMR mea- surements were used to determine the in-plane four-fold and uniaxial magnetic anisotropies, K₁ and K_u, and the effective demagnetizing field perpendicular to the film surface, 4␲^Meff, as a function of the film thickness d.³The magnetic anisotropies were well described by the bulk and interface magnetic properties, respectively.² The reproducible mag- netic anisotropies and small FMR linewidths provided an excellent opportunity for the investigation of nonlocal relax- ation processes in magnetic multilayer films. The thin Fe films which were studied in the single layer structures were regrown as a part of magnetic double layer structures. The thin Fe film共F1兲was separated from the second thick layer 共F2兲by a Au共001兲spacer共N兲of a variable thickness between 12 and 100 ML. The magnetic double layers were covered by a 20 ML Au共001兲capping layer. The thickness of the Au spacer layer was always smaller than the electron mean free path 共38 nm兲,⁴and hence allowed ballistic spin transfer be- tween the magnetic layers.

The interface magnetic anisotropies separated the FMR fields of F1 and F2 by a big margin (⬃1 kOe, see Fig. 1兲 allowing us to carry out FMR measurements in F1 with F2

a兲Author to whom correspondence should be addressed; electronic mail:

gwolters@sfu.ca

7545

0021-8979/2003/93(10)/7545/6/$20.00 © 2003 American Institute of Physics

possessing a small angle of precession compared to that in F1 and vice versa. The thin Fe film in the single and double layer structures had the same FMR field showing that the static interlayer exchange coupling¹ through the Au spacer was negligible.

The main results are as follows:共a兲The FMR linewidth in the thin films F1 always increased in the presence of a thick layer F2 and vice versa;共b兲The additional FMR line- width, ⌬H_add, followed an inverse dependence on the thin film thickness d₁;² and 共c兲 the additional FMR linewidth

⌬H_addin both the parallel共H in-plane兲and perpendicular共H perpendicular to the plane兲 FMR configuration was linearly dependent on the microwave frequency with no constant off- set. The additional Gilbert damping for the 16 ML thick film was found to be only weakly dependent on the crystallo- graphic direction, with the average value G_add⫽1.2⫻10⁸ s^⫺1. Its strength is comparable to the intrinsic Gilbert damp- ing in the single Fe film, 1.4⫻10⁸ s^⫺1.

THEORETICAL MODELS OF NONLOCAL DAMPING Berger⁵ evaluated the role of the s-d exchange interac- tion in magnetic double layers by allowing the magnetic mo- ment of one layer 共F1兲 to precess around the equilibrium direction while the other layer 共F2兲 was assumed to be sta- tionary, see the graphical representation in Fig. 2共a兲. Itinerant electrons entering the layer F1 through a sharp interface can- not immediately accommodate the direction of the precess- ing magnetization. Berger showed that this leads to an addi- tional exchange torque which is directed towards the magnetic equilibrium axis, and represents an additional re- laxation term. This relaxation torque is confined to a region near the F1/N interface whose thickness is given by the transverse spin relaxation coherence length L₀⫽␲^/(k↑

⫺k_↓), where k_↑ and k_↓ are the majority and minority Fermi k wave vectors in F1. L₀ is expected to be less than 1 nm.

The resulting relaxation torque in a magnetic double layer structure contributes to an additional interface FMR line- width ⌬H_add, such that

⌬H_add⬃共⌬␮⫹ប␻兲, 共2兲

where⌬␮⫽⌬␮_↑⫺⌬␮_↓ is the difference in the spin up and spin down Fermi level shifts, and␻ is the microwave angu- lar frequency. ⌬␮ is negligible for small angle precession, but can be brought in with ⫹ and⫺ sign by a dc current which is oriented perpendicular to interfaces.⁶The frequency dependent term in Eq. 共2兲 was obtained using the full dy- namic treatment of the s–d exchange interaction, and it is always positive.

Berger’s expression for the FMR linewidth, Eq.共2兲, was derived for a circular precession. One has to ask, what can be expected for the parallel FMR configuration where the de- magnetizing effect leads to a strong ellipticity in precessional motion. Berger⁷ included the contribution in Eq. 共2兲 to the nonlocal damping by using Slonczewski’s spin transport torque.⁶In this case the effective damping field for F1 can be written as

coef共⌬␮⫹ប␻兲c⫻M₁

M_s. 共3兲

where c is the direction of the magnetization in the stationary layer F2, and coef is a common prefactor. The vector product cÃM₁ in the effective field results in Bloch–Blombergen damping with the relaxation rate parameter proportional to the microwave frequency. In the perpendicular configuration Eq.共3兲results in the FMR linewidth which is strictly propor- tional to the microwave frequency共Gilbert-like兲, but for the

FIG. 1. The resonance fields at 24 GHz in the layer F1关16Fe, shown by (䊊)] and layer F2关40Fe, (쐓)] in 20Au/40Fe/16Au/16Fe/GaAs共001兲共the in- tegers represent the number of atomic layers兲as a function of the angle␸ between the applied field and the in-plane关100兴crystallographic direction.

A large in-plane uniaxial anisotropy field共0.5 kOe with the hard axis along 关110¯兴) in F1, is caused by dangling bonds of the GaAs共001兲substrate, leads to an accidental crossover at␸⫽115° and 150°. Notice that the resonance fields get locked together by the spin pumping effect at the accidental cross- over. Away from the crossover the resonance fields are separated by as much as ten FMR linewidths.

FIG. 2. An image representing the dynamic coupling between two magnetic layers which are separated by a nonmagnetic spacer N.共a兲represents two magnetic layers with different FMR fields. F1 is at resonance, and F2 is nearly stationary. A large gray arrow in the normal spacer describes the direction of the spin current. The dashed lines represent the instantaneous direction of the spin momentum. For small angle of precession they are nearly parallel to the transverse rf magnetization component shown in short solid arrows. F1 acts as a spin pump, F2 acts as a spin sink.共b兲represents a situation when F1 and F2 resonate at the same field. Both layers act as spin pumps and spin sinks. In this case the net spin momentum transfer across each interface is zero. No additional damping is present.

parallel configuration the FMR linewidth is proportional to

⬃(␻^/␥^)2/(B⫹H). In the parallel configuration the FMR linewidth is dependent on the ellipticity of precession.

Tserkovnyak et al.⁸ showed that the interface damping can be generated by a spin current from a ferromagnet 共F兲 into the adjacent normal metal 共NM兲 reservoirs. The spin current is generated by a precessing magnetic moment. A precessing magnetization at the F/NM interface acts as a

‘‘peristaltic spin-pump.’’ The direction of the spin current is perpendicular to the F/NM interface and points away from the interface towards the NM layer. The spin momentum which is carried away by the spin current is

j_spin⫽ ប 4␲^Arm⫻

dm

dt , 共4兲

where m is the a unit vector in the direction of M. The spin current can result共see below兲in magnetic damping. A_rfor F films thicker than L₀ is given by the scattering matrix ele- ments

A_r⫽1

2

兺

where r_mn^↑↓ are the reflection parameters at the NM/F inter- face for the spin up and down electrons. The sum in A_r is close to the number of the transverse channels in NM.⁹The sum is given by

A_r S ⫽ k_F²

4␲^{⫽0.85n2/3, 共6兲}

where S is the area of the interface, k_F is the Fermi wave vector, and n is the density of electrons per spin in NM.⁹ Brataas et al.^9,10 showed that A_r can be evaluated from the interface mixing conductance G_↑↓.¹¹ A_r⫽(h/e²) G_↑↓

⫽Sg_↑↓, where g_↑↓ represents ‘‘dimensionless interface mix- ing conductivity.’’

Now another important point has to be answered: ‘‘How is the generated spin current dissipated in the normal metal spacer N?’’ This answer can be found in Refs. 10 and 12.

These authors have shown that the transverse component of the spin current in N is entirely absorbed at the N/F2 inter- face 关see Fig. 2共a兲兴. For small precessional angles the spin current is almost entirely transverse. This means that the N/F2 interface acts as an ideal spin sink, and provides an effective spin brake for the precessing magnetic moment in F1. The spin momentum j_spinin the spin current has the form of Gilbert damping in F1. The Gilbert damping is given by the conservation of the total spin momentum

j_spin⫺1

␥

⳵^Mtot

⳵^{t ⫽0, 共7兲}

where M_totis the total magnetic moment in F1. After simple algebraical steps one obtains an expression for the dimen- sionless spin pump contribution␣sp to the damping

␣sp⫽ G_sp

␥^Ms⫽g␮B

g_↑↓

4␲^Ms

1

d₁, 共8兲

where d₁ is the thickness of F1, g_↑↓ is the dimensionless mixing conductivity, and G_sp is the spin pump Gilbert pa-

rameter. g is the electron g factor. The inverse dependence of

␣sp on the film thickness clearly testifies to its interfacial origin. The layers F1 and F2 act as mutual spin pumps and spin sinks. For small precessional angles the equation of mo- tion for F1 can be written as

1

␥

⳵^M1

⳵^{t ⫽⫺关M1⫻Heff,1兴⫹}

G₁

␥^2Ms

2

冋

册

⫹ ប 4␲^d1

g_↑↓,1m₁⫻⳵^m1

⳵^{t ⫺} ប 4␲^d1

g_↑↓,2m₂ ⳵^m2

⳵^{t ,} 共9兲 where M₁is the magnetization vector of F1, m_1,2are the unit vectors along M_1,2, and d₁ is the thicknesses of F1. The exchange of spin currents is a symmetric concept and the equation of motion for the layer F2 is obtained by inter- changing the indices 12.

The spin pump model is a rather exotic theory to those who are used to magnetic studies. One would expect that there is a direct connection to a more common concept which is applicable to magnetic multilayers. The obvious choice is interlayer exchange coupling. The interlayer ex- change interaction has been so far treated only in the static limit.¹³ One would expect that its dynamic part could create magnetic damping. A ferromagnetic sheet surrounded by a NM reservoir can be investigated by using a contact ex- change interaction between the ferromagnetic spins and the electrons in NM. A similar model was used by Yafet¹⁴ for calculating the static interlayer coupling. One can expand the linear response Kubo theory¹⁵for slow precessional motion using a linear approximation for a retarded magnetic moment

S共t⫺␶兲⬵S共t兲⫺␶⳵^S共t兲

⳵^{t , 共10兲}

where S(t) is the spin moment of the magnetic sheet at the instantaneous time t and ␶ is the time delay of the retarded response. The induced moment in NM at the F/NM interface results in an effective damping field which is proportional to the imaginary part of the rf transverse susceptibility of NM and the time derivative of the magnetic moment

H_damp^sd ⬃

冋

^冕

册

dM共t兲

dt . 共11兲 This damping term satisfies again the Gilbert phenomenol- ogy. By using the same interaction potential it is shown^16,28 that the Gilbert damping from the dynamic interlayer ex- change coupling, G_s⫺d, is similar to that using the spin- pumping theory⁸ combined with a perfect spin sink. This leads to an important conclusion: The spin pumping theory is equivalent to the dynamic response of the interlayer ex- change coupling. The rf susceptibility in Eq.共11兲allows one to account for electron–electron correlation effects in the normal metal. It has been shown^16,28that the Gilbert damp- ing is enhanced by the square of the Stoner factor S_E⫽关1

⫺UN(E_F)兴^⫺1,

G_s^enh_⫺d⫽G_s⫺dS_E², 共12兲

where U is the screened interatomic Coulomb interaction and N(E_F) the electron density of states, per atom, at the Fermi level in NM.

It is worthwhile to realize that the s–d exchange relax- ation mechanism also applies to bulk ferromagnets, and was evaluated by Heinrich et al.^17,18The Gilbert damping in this case is given by

G_s^bulk_⫺d⫽␹P

␶sf

, 共13兲

where ␹P is the Pauli susceptibility and ␶sf is the spin flip relaxation time of itinerant electrons in the ferromagnet. It should be noted that 1/␶sf in metals is proportional to the square of the spin orbit interaction.^17,18Using␹Pfrom Kries- man and Callen¹⁹ and ␶sf from the spin diffusion length in current perpendicular to plane giant magnetoresonance mea- surements one obtains for the bulk Gilbert damping G⫽5

⫻10⁶ and 1⫻10⁸ s^⫺1 for Co and permalloy 共Py兲, respec- tively, see the details in Ref. 18. This contribution is small in Co but it explains the intrinsic damping in Py. Fe is expected to behave like Co. The spin pumping mechanism is very effective for ultrathin films, but is negligible in bulk materi- als because its strength is inversely proportional to the thick- ness. Notice that the spin pumping mechanism does not have an explicit temperature dependence, while the bulk Gilbert damping 关see Eq. 共13兲兴, scales with 1/␶sf which is propor- tional to resistivity. One expects that there has to be an ad- ditional mechanism which depends explicitly on ␶sf. The origin of the interlayer exchange coupling lies in the itinerant nature of the electron carriers. It can be explained by using a spin dependent interface potential.²⁰ The effective field that acts on the layer F1 is given by differentiating the density of the interlayer exchange energy E^intwith respect to M₁

H_damp^int ⫽⫺⳵^Eint

⳵^M1

⫽⫺1

⍀

兺

These electrons are mostly confined to the N spacer. ⍀

⫽Sd₁ is the volume of the magnetic layer F1. The energy of electrons is dependent on the instantaneous orientation of the magnetic moments, and consequently the occupation number n_k,␴ of electronic states having energy ⑀k,␴ changes with time and this results in a ‘‘breathing Fermi surface.’’ This concept was also used in Refs. 21 and 22. However, this redistribution cannot be achieved instantaneously. The time lag between the instantaneous exchange field and the in- duced moment in the spacer is described by the transverse spin relaxation time ␶sf. In the limit of slow precessional motion the instantaneous electron distribution can be ap- proximated by

n_k,␴共t兲⫽n_k,␴关M₁共t兲兴⫺␶sf

⳵ⁿk,␴关M₁共t兲兴

⳵^{t , 共15兲} where n_k,␴关M₁(t)兴 is the static occupation number for the magnetic moment of the layer F1 with the magnetization along M₁(t). The first term in Eq. 共15兲 provides the static

interlayer exchange coupling field, and the second term pro- vides damping. The effective damping field can be evaluated by using Eqs.共14兲and共15兲:

H_damp^int ⫽␶sf

兺

冉

冊

共16兲 where the sum is carried out per unit area of F1. This effec- tive damping field is again proportional to the time derivative of the magnetic moment, and inversely proportional to the film thickness d; a clear indication of interface Gilbert damp- ing. However in this case the damping field is proportional to the spin relaxation time␶sf. Therefore this effect is explicitly dependent on the conductivity and represents a different con- tribution to the nonlocal damping compared to the spin pumping mechanism which is independent of ␶sf.

DISCUSSION OF THE RESULTS

Spin pumping and breathing Fermi surface theories pre- dict a Gilbert damping having a strictly linear dependence of

⌬H_addon the microwave frequency. Figure 3 shows that this is experimentally verified over a wide range of microwave frequencies. The dotted line represents the FMR linewidth calculated using Berger’s effective field 关see Eq. 共3兲兴. Sur- prisingly even in this case the measured microwave fre- quency dependence of⌬H_addis essentially linear. The differ- ence between the Gilbert damping and Berger’s damping is only apparent in the negative zero frequency offset共obtained by extrapolating the dotted line to zero microwave fre- quency兲. The fit using the the dotted line is obviously poorer than that using the straight line for Gilbert damping. The spin pumping theory is clearly the mechanism of preference for the nonlocal damping. Its validity can be tested by compar- ing calculations using Eq. 共9兲with the experimental results.

Figure 2 shows two extreme situations. In Fig. 2共a兲the FMR fields in F1 and F2 are separated by a big margin. In Fig.

2共b兲the FMR fields are the same. In共a兲one expects the full contribution from the nonlocal damping. ⌬H_addfor F1 and F2 should scale with their respective 1/d terms. In 共b兲 the

FIG. 3. The FMR linewidth for 16Fe共001兲as a function of the microwave frequency using (䊊) 20Au/16Fe/GaAs共001兲 single and (䊉)20Au/40Fe/

40Au/16Fe/GaAs共001兲double layer structure. (쐓) represent the additional part of the FMR linewidth⌬Haddin the double layer sample. The dotted line is a fit to the data obtained using Slonczewski’s effective damping 关see Eq.共3兲兴.

situation is symmetric, the net spin momentum flow through both interfaces is zero, and no additional damping is ex- pected. This behavior is well demonstrated in Fig. 4. The good agreement between theory and experiment clearly shows the validity of the spin pumping theory which is de- scribed by Eq. 共9兲. The magnetic layers even in the absence of static interlayer exchange coupling are coupled by the dynamic part of interlayer exchange. The spin sink effect at the N/F interface starts to be inefficient only when the N metal spacer thickness becomes comparable to the spin dif- fusion length. The spin diffusion length in Au is of the order of 100 nm. The static interlayer exchange coupling vanishes in our samples due to interface roughness on a length scale of a mere 10 ML共2 nm兲. One should point out that when the N metal spacer thickness starts to be comparable to the spin diffusion length then the N spacer on its own can act as an effective spin sink.^23,24

The quantitative comparison with predictions of the spin pumping theory is very favorable. First principles electron band calculations¹¹ resulted in g_↑↓⬇1.1⫻10¹⁵ cm^⫺2 for an alloyed Cu/Co共111兲 interface. By scaling this value to Au using Eq.共6兲one obtains G_sp⫽1.4⫻10⁸ s^⫺1 which is close to that measured by FMR. This is a surprising agreement considering the fact that calculations of the intrinsic damping in bulk metals have been carried out over the last three de- cades, and yet they have not been able to produce a compa- rable agreement with experiment.¹⁸

The breathing Fermi surface contribution to the Gilbert damping is proportional to the electron relaxation time␶sfof the N metal spacer 关see Eq. 共16兲兴. A test of the breathing Fermi surface contribution can be carried out by measuring the temperature dependence of the nonlocal damping. One expects proportionality with the sheet conductance (␶sf

⬃␶orb⬃␴) of the N spacer. The temperature dependence of the additional FMR linewidth, shown in Fig. 5, clearly indi-

cates that the strength of the breathing Fermi surface contri- bution is very small in Fe/Au/Fe共001兲. In fact, the observed temperature dependence of ⌬H_addis caused by the presence of spin dependent resistance in the Au spacer, which will be discussed in a separate article.

The dynamic exchange coupling theory 关see Eq. 共11兲兴, allows an enhancement of the additional Gilbert damping by the Stoner enhancement factor 关see Eq. 共12兲兴. In fact, our recent results using 20Au/4Pd/关Fe/Pd兴5/14Fe/GaAs共001兲 single and 20Au/40Fe/40Au/4Pd/关Fe/Pd兴5/14Fe/GaAs共001兲 double layer samples 共see Fig. 6兲, show some evidence for the Stoner enhancement factor. This structure incorporates a magnetic 关Fe/Pd兴5 superlattice with five repetitions of a 关1Fe/1Pd兴 unit cell. The N metal spacer is 4Pd40Au共001兲. Note that at ␸⫽135° the FMR linewidth is decreased down to the value which was observed for the single layer structure GaAs/14Fe关1Pd/1Fe兴5/4Pd/20Au共001兲. At ␸⫽135° the resonant fields in the 14Fe关1Pd/1Fe兴5 and 40Fe layers were almost identical, eliminating the nonlocal damping. The ad- ditional FMR linewidth along the cubic crystallographic axes (␸⫽0° and 90°) was enhanced by as much as a factor of 3 共see Fig. 6兲. The value of the nonlocal damping is signifi- cantly bigger than that expected from the simple spin pump-

FIG. 4. The FMR linewidth at 24 GHz as a function of the angle␸around the crossover of the FMR fields for 20Au/40Fe/14Au/16Fe/GaAs共001兲. The measured and calculated FMR signals were analyzed using two Lorenzian lineshapes. The Lorenzian peaks were characterized by their amplitudes, resonance fields and linewidths. The solid lines were obtained from calcu- lations using Eq.共9兲. The position of the FMR peaks is shown in Fig. 3. (䊉) correspond to F1共16Fe兲(䊊) correspond to F2共40 ML兲. Note that the FMR linewidth for the thinner sample, F1, first increases before it reaches its minimum value corresponding to its single 20Au/16Fe/GaAs共001兲 layer structure. Note also that the additional line broadening scales inversely with the film thickness.

FIG. 5. The additional FMR linewidth, ⌬Hadd, in 20Au/14Au/16Fe/

GaAs共001兲shown in black triangles, as a function of temperature. The tem- perature dependence of the sheet conductivity,␴, is shown in the dashed line. Note that the temperature dependence of⌬Haddis very weak.

FIG. 6. The dependence of the FMR linewidth in 14Fe关1Pd/1Fe兴5at 36 GHz as a function of the angle␸. (䊊) symbols correspond to the single layer measurements using a GaAs/14Fe关1Pd/1Fe兴5/4Pd/20Au共001兲structure, and (쐓) symbols correspond to the double layer measurements using a GaAs/

14Fe关1Pd/1Fe兴5/4Pd/40Au/40Fe/20Au共001兲structure.

ing mechanism. Metallic Pd is known to exhibit a strong Stoner enhancement in the dc susceptibility. These results clearly show that electron correlation effects in the N metal spacer have to be seriously considered.

It is interesting to explore the role of spin pumping in a bilayer 5Fe/12Cu/10Fe共001兲where the Fe layers are coupled by interlayer exchange energy. In this case one gets acoustic and optical precessional modes.¹ Calculations were carried out at 36 GHz using the spin pump and spin sink contribu- tions as shown in Eq.共9兲. For a moderate antiferromagnetic exchange coupling J⫽⫺0.2 ergs/cm², the optical peak is broadened by 200 Oe while the acoustic peak is only broad- ened by 36 Oe. For antiferromagnetic exchange coupling the optical peak mostly arises from the 5Fe layer. For zero inter- layer exchange coupling the spin pumping contribution to the FMR linewidth for the 5Fe layer is 150 Oe. This should be expected considering that the optical peak corresponds to an out of phase precession of the magnetic moments in the 5Fe and 10Fe layers, and therefore the spin momentum is more efficiently pumped. Experimentally, optical FMR peaks were always observed to be wider than the acoustic peaks. In a 5Fe/12Cu/10Fe sample grown on Ag共001兲 substrate the measured optical peak was broadened by 500 Oe.²⁵ The above calculation indicates that approximately 50% of the broadening was due to spin pumping and 50% was caused by an inhomogeneous exchange coupling.

CONCLUSIONS

We have shown that nonlocal damping by the transfer of spin momentum can be realized in magnetic multilayer films.

The effect is significant in ultrathin films. Theoretical models were presented for the nonlocal damping. It has been dem- onstrated that the nonlocal interface Gilbert damping in mag- netic multilayers is well described by the concept of spin pumps and spin sinks. It has been shown that this effect is directly related to the dynamics of the interlayer exchange coupling. By proper engineering of multilayer structures one can create magnetic damping which significantly surpasses that in the bulk materials.

ACKNOWLEDGMENTS

The authors thank Y. Tserkovnyak, A. Brataas, G. E. W.

Bauer, J. F. Cochran, and K. Myrtle for their assistance and

valuable discussions during the course of this work. Finan- cial support from the Natural Sciences and Engineering Re- search Council of Canada 共NSERC兲 and Canadian Institute for Advanced Research 共CIAR兲 is gratefully acknowledged.

G.W. thanks the German Academic Exchange Service 共DAAD兲for generous financial support.

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